1. Field of the Invention
The present invention relates to x-ray computed tomography (CT) and, more particularly, to systems and methods for theoretically exact interior reconstruction using compressive sampling (or compressed sensing or under another name for the essentially same thing) technology that is based an appropriate signal sparsity model, with the extension of such techniques to other tomographic modalities, such as PET/SPECT, MRI, and others that use imaging geometries of straight rays or nearly straight rays.
2. Background Description
One conventional wisdom is that the interior problem (exact reconstruction of an interior ROI only from data associated with lines through the ROI) does not have a unique solution (see F. Natterer, The mathematics of computerized tomography. Classics in applied mathematics 2001, Philadelphia: Society for Industrial and Applied Mathematics). Nevertheless, it is highly desirable to perform interior reconstruction for radiation dose reduction and other important reasons. Hence, over past years a number of image reconstruction algorithms were developed that use an increasingly less amount of raw data (D. L. Parker, Optimal short scan convolution reconstruction for fanbeam ct. Med. Phys., 1982. 9(2): p. 254-257; F. Noo, R. Clackdoyle, and J. D. Pack, A two-step hilbert transform method for 2d image reconstruction. Physics in Medicine and Biology, 2004. 49(17): p. 3903-3923; M. Defrise, et al., Truncated hilbert transform and image reconstruction from limited tomographic data. Inverse Problems, 2006. 22(3): p. 1037-1053). Specifically, motivated by the major needs in cardiac CT, CT guided procedures, nano-CT and so on (G. Wang, Y. B Ye, and H. Y Yu, Interior tomography and instant tomography by reconstruction from truncated limited-angle projection data, U.S. Pat. No. 7,697,658, Apr. 13, 2010), by analytic continuation we proved that the interior problem can be exactly and stably solved if a sub-region in an ROI within a field-of-view (FOV) is known (see Y. B. Ye, et al., A general local reconstruction approach based on a truncated hilbert transform. International Journal of Biomedical Imaging, 2007, Article ID: 63634, 8 pages; Y. B. Ye, H. Y. Yu, and G. Wang, Exact interior reconstruction with cone-beam CT. International Journal of Biomedical Imaging, 2007, Article ID: 10693, 5 pages; Y. B. Ye, H. Y. Yu, and G. Wang, Exact interior reconstruction from truncated limited-angle projection data. International Journal of Biomedical Imaging, 2008 ID: 427989, 6 Pages; H. Y. Yu, Y. B. Ye, and G. Wang, Local reconstruction using the truncated hilbert transform via singular value decomposition. Journal of X-Ray Science and Technology, 2008. 16(4): p. 243-251). Similar results were also independently reported by others (see H. Kudo, et al., Tiny a priori knowledge solves the interior problem in computed tomography. Phys. Med. Biol., 2008. 53(9): p. 2207-2231; M. Courdurier, et al., Solving the interior problem of computed tomography using a priori knowledge. Inverse Problems, 2008, Article ID 065001, 27 pages.). Although the CT numbers of certain sub-regions such as air in a trachea and blood in an aorta can be indeed assumed, how to obtain precise knowledge of a sub-region generally can be difficult in some cases such as in studies on rare fossils or certain biomedical structures. Therefore, it would be very valuable to develop more powerful interior tomography techniques.
Another conventional wisdom is that data acquisition should be based on the Nyquist sampling theory, which states that to reconstruct a band-limited signal or image, the sampling rate must at least double the highest frequency of nonzero magnitude. Very interestingly, an alternative theory of compressive sampling (or compressed sensing or under another name for the essentially same thing) (CS) has recently emerged which shows that high-quality signals and images can be reconstructed from far fewer data/measurements than what is usually considered necessary according to the Nyquist sampling theory (see D. L. Donoho, Compressed sensing. IEEE Transactions on Information Theory, 2006. 52(4): p. 1289-1306; E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006. 52(2): p. 489-509). The main idea of CS is that most signals are sparse in an appropriate domain (an orthonormal system or more generally a frame system), that is, a majority of their coefficients are close or equal to zero, when represented in that domain. Typically, CS starts with taking a limited amount of samples in a much less correlated basis, and the signal is exactly recovered with an overwhelming probability from the limited data via the L1 norm minimization (or minimization of another appropriate norm). Since samples are limited, the task of recovering the image would involve solving an underdetermined matrix equation, that is, there is a huge amount of candidate images that can all fit the limited measurements effectively. Thus, some additional constraint is needed to select the “best” candidate. While the classical solution to such problems is to minimize the L2 norm, the recent CS results showed that finding the candidate with the minimum L1 norm, also basically equivalent to the total variation (TV) minimization in some cases (L. L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D, 1992. 60(1-4): p. 259-268), is a reasonable choice, and can be expressed as a linear program and solved efficiently using existing methods (see D. L. Donoho, Compressed sensing. IEEE Transactions on Information Theory, 2006. 52(4): p. 1289-1306; E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006. 52(2): p. 489-509).
Because the x-ray attenuation coefficient often varies mildly within an anatomical component, and large changes are usually confined around borders of tissue structures, the discrete gradient transform (DGT) has been widely utilized as a sparsifying operator in CS-inspired CT reconstruction (for example, E. Y. Sidky, C. M. Kao, and X. H. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct. Journal of X-Ray Science and Technology, 2006. 14(2): p. 119-139; G. H. Chen, J. Tang, and S. Leng, Prior image constrained compressed sensing (piccs): A method to accurately reconstruct dynamic ct images from highly undersampled projection data sets. Medical Physics, 2008. 35(2): p. 660-663; H. Y. Yu, and G. Wang, Compressed sensing based interior tomography. Phys Med Biol, 2009. 54(9): p. 2791-2805; J. Tang, B. E. Nett, and G. H. Chen, Performance comparison between total variation (tv)-based compressed sensing and statistical iterative reconstruction algorithms. Physics in Medicine and Biology, 2009. 54(19): p. 5781-5804). This kind of algorithms can be divided into two major steps. In the first step, an iteration formula (e.g. SART) is used to update a reconstructed image for data discrepancy reduction. In the second step, a search method (e.g. the standard steepest descent technique) is used in an iterative framework for TV minimization. These two steps need to be iteratively performed in an alternating manner. However, there are no standard stopping and parameter selection criteria for the second step. Usually, these practical issues are addressed in an ad hoc fashion.
On the other hand, soft-threshold nonlinear filtering (see M. A. T. Figueiredo, and R. D. Nowak, An em algorithm for wavelet-based image restoration. IEEE Transactions on Image Processing, 2003. 12(8): p. 906-916; I. M. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 2004. 57(11): p. 1413-1457; I. M. Daubechies, M. Fornasier, and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints. Journal of Fourier Analysis and Applications, 2008. 14(5-6): p. 764-792) was proved to be a convergent and efficient algorithm for the norm minimization regularized by a sparsity constraint. Unfortunately, because the DGT is not invertible, it does not satisfy the restricted isometry property (RIP) required by the CS theory (see D. L. Donoho, Compressed sensing. IEEE Transactions on Information Theory, 2006. 52(4): p. 1289-1306; E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006. 52(2): p. 489-509) and soft-threshold algorithm (see I. M. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 2004. 57(11): p. 1413-1457; I. M. Daubechies, M. Fornasier, and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints. Journal of Fourier Analysis and Applications, 2008. 14(5-6): p. 764-792). In other words, the soft-threshold algorithm cannot be directly applied for TV minimization. The above problem can be overcome using an invertible sparsifying transform such as a wavelet transform for image compression. For an object of interest such as a medical image, we can find an orthonormal basis (or more generally, a frame) to make the object sparse in terms of significant transform coefficients. Then, we can perform image reconstruction from a limited number of non-truncated or truncated projections by minimizing the corresponding L1 norm.
Inspired by the CS theory, we proved and demonstrated the interior tomography is feasible in the CS framework assuming a sparsity model; specifically a piecewise-constant image model and a number of high-order models (see H. Y. Yu, and G. Wang, Compressed sensing based interior tomography. Phys Med Biol, 2009. 54(9): p. 2791-2805; H. Y. Yu, et al., Supplemental analysis on compressed sensing based interior tomography. Phys Med Biol, 2009. 54(18): p. N425-N432; W. M. Han, H. Y. Yu and G. Wang; A general total variation minimization theorem for compressed sensing based interior tomography; International Journal of Biomedical Imaging, Article ID: 125871, 2009, 3 pages; H. Y. Yu, et al; Compressive sampling based interior tomography for dynamic carbon nanotube Micro-CT; Journal of X-ray Science and Technology, 17(4): 295-303, 2009). The above finding has been extended to interior SPECT reconstruction assuming a piecewise-polynominial image model and introducing a high-order total variation concept (H. Y. Yu, J. S. Yang, M. Jiang, G. Wang: Methods for Exact and Approximate SPECT/PET Interior Reconstruction. VTIP No.: 08-120, U.S. Patent Application No. 61/257,443, Date Filed: Nov. 2, 2009), CT is a special case of SPECT when the attenuation background is negligible (J. S. Yang, H. Y. Yu, M. Jiang and G. Wang; High order total variation minimization for interior tomography, Inverse Problems, 26(3), Article ID: 035013, 2010, 29 pages).
Based on the recent mathematical findings made by Daubechies et al., we adapted a simultaneous algebraic reconstruction technique (SART) for image reconstruction from a limited number of projections subject to a sparsity constraint in terms of an invertible sparsifying transform (H. Y. Yu and G. Wang: SART-type image reconstruction from a limited number of projections with the sparsity constraint; International Journal of Biomedical Imaging, Article ID: 934847, 2010, to appear), and constructed two pseudo-inverse transforms of un-invertible transforms for the soft-threshold filtering (H. Y. Yu and G. Wang: Soft-threshold filtering approach for reconstruction from a limited number of projections, Physics in Medicine and Biology, pending revision).